{"id":1118,"date":"2021-01-12T13:44:39","date_gmt":"2021-01-12T21:44:39","guid":{"rendered":"http:\/\/theothermath.com\/?p=1118"},"modified":"2025-10-28T10:43:06","modified_gmt":"2025-10-28T17:43:06","slug":"chip-model-for-multiplication","status":"publish","type":"post","link":"https:\/\/theothermath.com\/index.php\/2021\/01\/12\/chip-model-for-multiplication\/","title":{"rendered":"Chip model for multiplication"},"content":{"rendered":"<p>This is the second part of a series of posts showing how to visually represent the four operations of whole numbers using the CHIP MODEL. The view the first part in which we show how to model addition and subtraction, go <a href=\"https:\/\/theothermath.com\/index.php\/2021\/01\/05\/chip-model-for-addition-and-subtraction\/\"><strong>here<\/strong><\/a>.<\/p>\n<hr \/>\n<p class=\"c0\"><span class=\"c17\">\u00a0<\/span><\/p>\n<p class=\"c0\"><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/YFA5VC0GFPs\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p class=\"c0\"><span class=\"c17\">Just as the chip model is a particularly effective way in visually represent addition and subtraction, we can also visually model multiplication with the chip model.<\/span><\/p>\n<p class=\"c0\"><span class=\"c3\">There are a variety to think of multiplication such as skip counting, area model, and repeated addition. With the chip model, however, we will rely on thinking of multiplication as \u201cequal groups\u201d.<\/span><\/p>\n<p class=\"c0\"><span class=\"c3\">34 x 7 means \u201c34 groups of 7\u201d and also \u201c7 groups of 34\u201d. <\/span><\/p>\n<p class=\"c0\"><span class=\"c3\">Most people would think it is easier to model 34 x 7 as \u201c7 groups of 34\u201d.<\/span><\/p>\n<table class=\"c19\">\n<tbody>\n<tr class=\"c5\">\n<td class=\"c7\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><span class=\"c3\">We begin by modeling 34 on the place value chart.<\/span><\/p>\n<\/td>\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><img title=\"\" src=\"https:\/\/lh6.googleusercontent.com\/vKNSAYIZpnid3ZUKgiVm51aouvTPT3Ch6Wh19ElH2_-7srumMafr48RYKnHypI_9BUvNwmvjuK3KkdP1CYvETu_J7gBh2dPhB0AhBq1U-aJaspSIXavv5y4p-033MbPWuQ\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<tr class=\"c5\">\n<td class=\"c7\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><span class=\"c3\">Then we model seven groups of 34.<\/span><\/p>\n<p class=\"c2\"><span class=\"c3\">At this point, we can see that the unsimplified product is<\/span><\/p>\n<p class=\"c11\"><span class=\"c3\">21 tens and 28 ones<\/span><\/p>\n<\/td>\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><img title=\"\" src=\"https:\/\/lh5.googleusercontent.com\/HM0DF6u01SzxsKY93j_OvoJy8PYd7puF71y2Hxa5qotB-c3XDG5EyDgZjYFzMTaqfE7PpRYGayYq_5ooBirUtsEgHJ89qhBugZYQnJ4to8AC1b6RfdZ6FGm59gHFGVYerA\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<tr class=\"c5\">\n<td class=\"c7\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\">Now we look for opportunities to do the exchange rate of <em><span class=\"c8\">10 littles equals 1 big<\/span><\/em><span class=\"c3\">.<\/span><\/p>\n<p class=\"c2\"><span class=\"c3\">We see there is a multiple opportunities to do so because there are more than 10 dots in both the tens column and the ones column. It does not matter which exchanges we do first.<\/span><\/p>\n<\/td>\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><img title=\"\" src=\"https:\/\/lh5.googleusercontent.com\/FEqrrLCdbqpmdFDVzDGWI95-7FE3QxKacNVG7c9U8Utr8g3XGuR1JL23ReSFsXCM4ZIWH5G2xGjlRUFDiIE0TjZL-jFRrl2wIulJQJgMc8LjxyierWXeXsWwSDb-zb8wsw\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<tr class=\"c5\">\n<td class=\"c7\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\">After completing all the possible exchanges of\u00a0<em><span class=\"c8\">10 littles equals 1 big<\/span><\/em><span class=\"c3\">, we count the remaining dots in each column.<\/span><\/p>\n<p class=\"c11\"><span class=\"c3\">2 hundreds and 3 tens and 8 ones<\/span><\/p>\n<p class=\"c2\"><span class=\"c3\">The product is 238.<\/span><\/p>\n<p class=\"c1\">\n<\/td>\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><img title=\"\" src=\"https:\/\/lh4.googleusercontent.com\/I1z9flXsGNwyO9nIF_18HFOLN7e3DUG1i4uuuycgPWLOhRKsiuDvLqC7XUJoFqpdiH4ZahGp8zLrrm050GF603KZiw0X4ehqJsInkjOo63eUaq5ayU2haIXk-ZVUjtAa8Q\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"c0\"><span class=\"c3\">Let\u2019s do a slightly larger example of a 3-digit number multiplied by a 1-digit number.<\/span><\/p>\n<p class=\"c0\"><span class=\"c3\">137 x 5 will be modeled as \u201c5 groups of 137\u201d<\/span><\/p>\n<table class=\"c19\">\n<tbody>\n<tr class=\"c5\">\n<td class=\"c7\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><span class=\"c3\">We begin by modeling 137 on the place value chart.<\/span><\/p>\n<\/td>\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><img title=\"\" src=\"https:\/\/lh6.googleusercontent.com\/YRqQyJkSkd5hl30Ejee3OhJBnXUSiWErgLEWGqP-iO5V9zuosz_VxY0dAfVaLRFRrARohY6RLAQW8VRjogfxIH0Yz5iUgMzpimTWcEdrXuX7dbaW7UVzJGXIaPUnNbdATw\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<tr class=\"c5\">\n<td class=\"c7\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><span class=\"c3\">Then we model five groups of 137.<\/span><\/p>\n<p class=\"c2\"><span class=\"c3\">At this point, we can see that the unsimplified product is<\/span><\/p>\n<p class=\"c11\"><span class=\"c3\">5 hundreds and 15 tens and 35 ones<\/span><\/p>\n<\/td>\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><img title=\"\" src=\"https:\/\/lh6.googleusercontent.com\/4UotuF4bjq5CQ2N5h2KVAlF7itxDZH909qWiRfqECzLBv9yCKAULP8Gytgruec4tZb3wavnMvyDeshL9Ca6o62M2cn7iUkIhr2WQmm5y07DeuRKcKXDkrjCQvpy85Ff-SA\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<tr class=\"c5\">\n<td class=\"c7\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\">Now we look for opportunities to do the exchange rate of <em><span class=\"c8\">10 littles equals 1 big<\/span><\/em><span class=\"c3\">.<\/span><\/p>\n<p class=\"c2\"><span class=\"c3\">We see there is a four opportunities to do so.<\/span><\/p>\n<\/td>\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><img title=\"\" src=\"https:\/\/lh5.googleusercontent.com\/vDmPkM3qRB5EV4xdbXsdcUQ5u8Zf-ih97fMIKV3AdG63RMtX0r0P2uYyR_hzeN-lMrX253x7CWCHUo0dhpN5_DUw0lVfdeRooF9ECHZ484ZanhuWEjpdvkzHpmrf2fBrhA\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<tr class=\"c5\">\n<td class=\"c7\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\">After completing all the possible exchanges of <em><span class=\"c8\">10 littles equals 1 big<\/span><\/em><span class=\"c3\">, we count the remaining dots in each column.<\/span><\/p>\n<p class=\"c11\"><span class=\"c3\">6 hundreds and 8 tens and 5 ones<\/span><\/p>\n<p class=\"c2\"><span class=\"c3\">The product is 685.<\/span><\/p>\n<p class=\"c1\">\n<\/td>\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><img title=\"\" src=\"https:\/\/lh3.googleusercontent.com\/IvJdwyj2QdwGf3vrN-iGKYcOoZVc7jv-Wm6MzAUrA_csYAfYrPosSZvffi8xDIxd3PKcOf7Fx4CCa6qLqD91GrxqgUZGaUOZ1h7MhVga5F-gTlniRu29d4ABBWuSpeoaVA\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"c0\"><span class=\"c3\">Eventually, students will show the chip model and the algorithm side-by-side, using the chip model one the left and recording each step with numbers on the right side.<\/span><\/p>\n<p class=\"c0\"><img title=\"\" src=\"https:\/\/lh5.googleusercontent.com\/pE02s1K8n6M5dlShfEHFI3BX3DREDGBZX50m3Bx5fX2jxKlgH9ill8ALdNcNv5o3Xxx6GKylc3LEwMQzjzgEiAnUT-9KwbIrMXCyYrLXJRpvdpTVsXYRNCecxmLuRvSrfQ\" alt=\"\" \/><\/p>\n<p class=\"c0\"><span class=\"c3\">For our final multiplication example, we will explore a possible technique for efficiently modeling the chips when there is a large number of groups we need to draw on the place value chart.<\/span><\/p>\n<p class=\"c0\"><span class=\"c3\">For example, 24 x 14 would require drawing either 24 groups of 14 or 14 groups of 24, both of which would be a lot of dots to draw!<\/span><\/p>\n<table class=\"c19\">\n<tbody>\n<tr class=\"c5\">\n<td class=\"c7\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><span class=\"c3\">Let\u2019s model 24 x 14 as 14 groups of 24, so we will begin by modeling 137 on the place value chart.<\/span><\/p>\n<\/td>\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><img title=\"\" src=\"https:\/\/lh3.googleusercontent.com\/dGJsTe-JuBzBMbKvhZuBMwEJOTOgx8wGYlnbRAG3x6qR83fDKFvWpCFvaVz3KU5F7cWQSWqwd7K1f2fOwm4AeIiymPY7iTPWb59veJHn2DPfv44o-Io1U_xe7byED9JwDQ\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<tr class=\"c5\">\n<td class=\"c7\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><span class=\"c3\">To efficiently model 14 copies of a dot in any column of the place value chart, we use a line to represent 10 copies of that dot and then an additional 4 dots.<\/span><\/p>\n<p class=\"c2\"><span class=\"c3\">We do this for each dot representing 24.<\/span><\/p>\n<p class=\"c2\"><span class=\"c3\">At this point, we can see that the unsimplified product is<\/span><\/p>\n<p class=\"c11\"><span class=\"c3\">28 tens and 56 ones<\/span><\/p>\n<\/td>\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><img title=\"\" src=\"https:\/\/lh6.googleusercontent.com\/WOw_Wsk85kfKqGAnq286-Geb8osfxDXo-Liv25JT7Gpt2onH0-0Vi6bJOpbWWtHcSfPn3zHwlBGYNV1QX9heftUrg5RSCcR5AsiR5kmYmeolqYwjhHIIjO-xzTDpaUuKZg\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<tr class=\"c5\">\n<td class=\"c7\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\">Now we look for opportunities to do the exchange rate of <em><span class=\"c8\">10 littles equals 1 big<\/span><\/em><span class=\"c3\">.<\/span><\/p>\n<p class=\"c2\">Each line represents 10 dots, so we begin by exchanging each line (10 dots) with a dot in the column to the left. This is the same <em><span class=\"c8\">10 littles equals 1 big<\/span><\/em><span class=\"c3\">\u00a0that we have been doing in all the previous examples.<\/span><\/p>\n<\/td>\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><img title=\"\" src=\"https:\/\/lh4.googleusercontent.com\/xKOl0KYQ65svYBCysdo6a9JmcUloG18b2Mb0ax0rYwWPRqyIOiSWkKBUXik9gdAiLk2eMDVRM3Y0aAssBKy2WreL0mzBOTkWsOckIyjjnp5nNKjgmth-PDWQ6iqm475U3g\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<tr class=\"c5\">\n<td class=\"c7\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\">After completing all the possible exchanges of <em><span class=\"c8\">10 littles equals 1 big<\/span><\/em><span class=\"c3\">, we count the remaining dots in each column.<\/span><\/p>\n<p class=\"c11\"><span class=\"c3\">3 hundreds and 3 tens and 6 ones<\/span><\/p>\n<p class=\"c2\"><span class=\"c3\">The product is 336.<\/span><\/p>\n<p class=\"c1\">\n<\/td>\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c2\"><img title=\"\" src=\"https:\/\/lh3.googleusercontent.com\/jwOF_OwDDHXnsgRt80VFPskDdG0Z_1Kct4gHVYiQXfakoz_DWPwQXHJUBflKaY5SABZAfVdVNkjgW69PeIlUYb1DX2q1amymkaQUaPhTqlGPgEq0I9H0F3dVb6dun77wFg\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"c5\"><span class=\"c1\">The chip model can lead to the partial products method if we consider a different way to simplify all the lines and dots in the place value chart.<\/span><\/p>\n<table class=\"c6\">\n<tbody>\n<tr class=\"c11\">\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c0\"><span class=\"c1\">Let\u2019s revisit the visual model after representing 14 groups of 24.<\/span><\/p>\n<\/td>\n<td class=\"c15\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c0\"><img loading=\"lazy\" class=\"\" title=\"\" src=\"https:\/\/lh6.googleusercontent.com\/WOw_Wsk85kfKqGAnq286-Geb8osfxDXo-Liv25JT7Gpt2onH0-0Vi6bJOpbWWtHcSfPn3zHwlBGYNV1QX9heftUrg5RSCcR5AsiR5kmYmeolqYwjhHIIjO-xzTDpaUuKZg\" alt=\"\" width=\"444\" height=\"222\" \/><\/p>\n<\/td>\n<\/tr>\n<tr class=\"c11\">\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c0\"><span class=\"c1\">Now we consider the lines and dots in four parts.<\/span><\/p>\n<\/td>\n<td class=\"c15\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c0\"><img loading=\"lazy\" class=\"\" title=\"\" src=\"https:\/\/lh5.googleusercontent.com\/3R_c4DMriOh1958ZOFp5NP5zOo3Ba3CZAOhmlX2cTPpllJfPFxfk34DF087OvGciJ_gr37qqQeCRuASYzmqofXAFwxJJSuayBy5uyAHHFtUC0enD87ztfoRFFO38LgYOZg\" alt=\"\" width=\"403\" height=\"207\" \/><\/p>\n<\/td>\n<\/tr>\n<tr class=\"c11\">\n<td class=\"c9\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c0\"><span class=\"c1\">Let\u2019s interpret the meaning and value of each of the four parts:<\/span><\/p>\n<p class=\"c0\"><span class=\"c1\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Upper left: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a020 tens = 200<\/span><\/p>\n<p class=\"c0\"><span class=\"c1\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Upper right: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a040 ones = 40<\/span><\/p>\n<p class=\"c0\"><span class=\"c1\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Lower left: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a08 tens = 80<\/span><\/p>\n<p class=\"c0\"><span class=\"c1\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Lower right: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a016 ones = 16<\/span><\/p>\n<p class=\"c0\"><span class=\"c1\">Then we add the four partial products to find the product of 336.<\/span><\/p>\n<\/td>\n<td class=\"c15\" colspan=\"1\" rowspan=\"1\">\n<p class=\"c0\"><img loading=\"lazy\" class=\"\" title=\"\" src=\"https:\/\/lh6.googleusercontent.com\/aKgq8DTthHNftClll5vOmZDdgbwCR4RyQxyXer516qQ-rbdZGxy4lNOia2wQVfiD6bsi5--8p-qrfpFl5WdAVhBeTKgMNzsZqJnzf9zUue2YOkvKNMEw67D6IEQBlYCFIw\" alt=\"\" width=\"384\" height=\"196\" \/><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p class=\"c5\"><span class=\"c1\">To summarize this section on multiplication, there are two things for students to know:<\/span><\/p>\n<ol class=\"c20 lst-kix_h9727ubxpqfn-0 start\" start=\"1\">\n<li class=\"c5 c13 li-bullet-0\">Multiplication means <span class=\"c18 c10\">equal groups<\/span><\/li>\n<li class=\"c5 c13 li-bullet-0\">The exchange rate is <em>10 littles<span class=\"c1\">\u00a0equals 1 big<\/span><\/em><\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is the second part of a series of posts showing how to visually represent the four operations of whole numbers using the CHIP MODEL. The view the first part in which we show how to model addition and subtraction, go here. \u00a0 Just as the chip model is a particularly effective way in visually [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":1120,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[4],"tags":[29,28,56,42],"_links":{"self":[{"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/posts\/1118"}],"collection":[{"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/comments?post=1118"}],"version-history":[{"count":5,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/posts\/1118\/revisions"}],"predecessor-version":[{"id":1125,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/posts\/1118\/revisions\/1125"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/media\/1120"}],"wp:attachment":[{"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/media?parent=1118"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/categories?post=1118"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theothermath.com\/index.php\/wp-json\/wp\/v2\/tags?post=1118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}